We examine the infinitesimal period relations of a 3-fold with trivial canonical bundle and show that they represent an exterior differential system on the higher period domain that is the first prolongation of a contact system on a `de-prolonged' period domain. This has some interesting consequences for relations between the so called A-periods and B-periods, showing that they can be expressed in terms of a single `potential' function.
Of course, in recent years, 3-folds with trivial canonical bundle have become a central part of string theory. The literature on these complex manifolds and their geometry and deformation theory has seen an explosive growth, which I am not competent to catalog. I would suggest that you contact my colleague, D. Morrison (email@example.com), if you want pointers to a survey article.
Reprints are available.